1. Field of the Invention
The present invention relates to operational transconductance amplifiers widely used to implement analog circuits such as filters, and more particularly, to an operational transconductance amplifier (OTA) for a Gm-C filter fabricated by an integrated circuit (IC) process.
This work was supported by the IT R&D program of Ministry of Information and Communication/Institute for Information Technology Advancement [2005-S-075-02, Development of SoC for Wired and Wireless Unified Network.]
2. Discussion of Related Art
In a conventional wireless communication transceiver, a high frequency analog signal at radio frequency (RF) stage is converted to a low frequency analog signal through a low-noise amplifying and downconverting process. The resultant analog signal generally contains unwanted frequency signal components. A filter eliminates such frequency components and passes only a pure signal component to a digital signal processing block. A Gm-C filter includes an operational transconductance amplifier (OTA) as a basic building block.
The OTA is a functional block which receives a voltage signal as an input and outputs a current signal. Assuming that an input signal is Vinput and an output signal is Ioutput there exists the following relationship between the input and output signal:Ioutput=gmVinput  (1)where gm denotes a proportional coefficient commonly called a conductance gain or simply conductance. An LC filter typically requires an inductor and a capacitor of large value. So, it is difficult for the inductor to be fabricated on an integrated circuit because of occupied area or the quality factor. Accordingly, an active filter is used as an alternative to the LC filter.
In the active filter, an impedance of the inductor is implemented by an integrator or a gyrator, which consists of a transconductance element and a capacitor. A functional block, called an impedance inverter, which allows a capacitor to act as an inductor can be implemented by a gyrator consisting of two operational transconductance amplifiers.
But, we encounter various nonideal or nonlinear characteristics while we design OTA. Only after you solve those problems can you implement the desired operational characteristics of the filter. In particular, in order to implement a filter operating at a high frequency, it is desirable to minimize number of OTA internal nodes causing parasite capacitances, which is the most serious obstacle to the high frequency operation of filters.
A representative example that adopts this idea is an operational transconductance amplifier proposed by Bram Nauta (“Nauta OTA”). In general, the Nauta OTA includes single input single output inverter as basic unit cell, as shown in FIG. 1. The Nauta OTA effectively amplifies only the differential mode component among signal components while suppressing the common mode component. Since the inverter, which has no nodes other than power and ground nodes, is used as the basic unit cell, the Nauta OTA is highly advantageous for high frequency operation. And, since the Nauta OTA includes only two transistors in series between the power node and the ground node, it is also suitable for a low voltage operation.
However, in the Nauta OTA having such a structure, a common mode component of an input signal appears with a certain gain at an output side, and it is difficult to increase the quality factor. These will now be analyzed in conjunction with the conventional Nauta OTA shown in FIG. 1, where
                              i          1                =                                            g                              m                ⁢                                                                  ⁢                1                                      ⁢                          v                              i                ⁢                                                                  ⁢                1                                              +                                    g                              m                ⁢                                                                  ⁢                2                ⁢                a                                      ⁢                          v                              o                ⁢                                                                  ⁢                1                                              +                                    g                              m                ⁢                                                                  ⁢                2                ⁢                b                                      ⁢                          v                              o                ⁢                                                                  ⁢                2                                              +                                    (                                                g                                      o                    ⁢                                                                                  ⁢                    1                                                  +                                  g                                      o                    ⁢                                                                                  ⁢                    2                    ⁢                    a                                                  +                                  g                                      o                    ⁢                                                                                  ⁢                    2                    ⁢                    b                                                              )                        ⁢                          v                              o                ⁢                                                                  ⁢                1                                                                        (        2        )            and
                              i          2                =                                            g                              m                ⁢                                                                  ⁢                1                                      ⁢                          v                              i                ⁢                                                                  ⁢                2                                              +                                    g                              m                ⁢                                                                  ⁢                2                ⁢                a                                      ⁢                          v                              o                ⁢                                                                  ⁢                2                                              +                                    g                              m                ⁢                                                                  ⁢                2                ⁢                b                                      ⁢                          v                              o                ⁢                                                                  ⁢                1                                              +                                    (                                                g                                      o                    ⁢                                                                                  ⁢                    1                                                  +                                  g                                      o                    ⁢                                                                                  ⁢                    2                    ⁢                    a                                                  +                                  g                                      o                    ⁢                                                                                  ⁢                    2                    ⁢                    b                                                              )                        ⁢                          v                              o                ⁢                                                                  ⁢                2                                                                        (        3        )            
As usual, assuming (gm1=gm2a=gm2b)=gm and
(go1=go2a=go2b)=go, we obtain Equations 4 and 5:i1=gm(vi1+vo1+vo2)+3govo1  (4)andi2=gm(vi2+vo2+vo1)+3govo2  (5)
Here,vo1=−zi1 vo2=−zi2  (6)
And, Equations 7 and 8 are obtained:vo1=−zgm(vi1+vo1+vo2)−3zgovo1  (7)vo2=−zgm(vi2+vo2+vo1)−3zgovo2  (8)
These equations are expressed by a matrix equation:
                                          [                                                                                1                    +                                          zg                      m                                        +                                          3                      ⁢                                              zg                        o                                                                                                                                  zg                    m                                                                                                                    zg                    m                                                                                        1                    +                                          zg                      m                                        +                                          3                      ⁢                                              zg                        o                                                                                                                  ]                    ⁡                      [                                                                                v                                          o                      ⁢                                                                                          ⁢                      1                                                                                                                                        v                                          o                      ⁢                                                                                          ⁢                      2                                                                                            ]                          =                  -                                                    zg                m                            ⁡                              [                                                                            1                                                              0                                                                                                  0                                                              1                                                                      ]                                      ⁡                          [                                                                                          v                                              i                        ⁢                                                                                                  ⁢                        1                                                                                                                                                        v                                              i                        ⁢                                                                                                  ⁢                        2                                                                                                        ]                                                          (        9        )            
Both sides of Equation 9 are multiplied by y to obtain Equation 10:
                                          [                                                                                y                    +                                          g                      m                                        +                                          3                      ⁢                                              g                        o                                                                                                                                  g                    m                                                                                                                    g                    m                                                                                        y                    +                                          g                      m                                        +                                          3                      ⁢                                              g                        o                                                                                                                  ]                    ⁡                      [                                                                                v                                          o                      ⁢                                                                                          ⁢                      1                                                                                                                                        v                                          o                      ⁢                                                                                          ⁢                      2                                                                                            ]                          =                  -                                                    g                m                            ⁡                              [                                                                            1                                                              0                                                                                                  0                                                              1                                                                      ]                                      ⁡                          [                                                                                          v                                              i                        ⁢                                                                                                  ⁢                        1                                                                                                                                                        v                                              i                        ⁢                                                                                                  ⁢                        2                                                                                                        ]                                                          (        10        )            
This is a basic input/output equation of the conventional Nauta OTA.
Using a capacitive load (a capacitor) as load admittance, Equation 10 is Laplace transformed into Equation 11:
                                          [                                                                                sC                    +                                          g                      m                                        +                                          3                      ⁢                                              g                        o                                                                                                                                  g                    m                                                                                                                    g                    m                                                                                        sC                    +                                          g                      m                                        +                                          3                      ⁢                                              g                        o                                                                                                                  ]                    ⁡                      [                                                                                v                                          o                      ⁢                                                                                          ⁢                      1                                                                                                                                        v                                          o                      ⁢                                                                                          ⁢                      2                                                                                            ]                          =                                            g              m                        ⁡                          [                                                                    1                                                        0                                                                                        0                                                        1                                                              ]                                ⁡                      [                                                                                v                                          i                      ⁢                                                                                          ⁢                      1                                                                                                                                        v                                          i                      ⁢                                                                                          ⁢                      2                                                                                            ]                                              (        11        )            
Both sides of Equation 11 are divided by C to obtain Equations 12 and 13:
                                          [                                                                                s                    +                                                                                            g                          m                                                ⁢                        3                        ⁢                                                  g                          o                                                                    C                                                                                                                                  g                      m                                        C                                                                                                                                          g                      m                                        C                                                                                        s                    +                                                                                            g                          m                                                +                                                  3                          ⁢                                                      g                            o                                                                                              C                                                                                            ]                    ⁡                      [                                                                                V                                          o                      ⁢                                                                                          ⁢                      1                                                                                                                                        V                                          o                      ⁢                                                                                          ⁢                      2                                                                                            ]                          =                  -                                                                      g                  m                                C                            ⁡                              [                                                                            1                                                              0                                                                                                  0                                                              1                                                                      ]                                      ⁡                          [                                                                                          V                                              i                        ⁢                                                                                                  ⁢                        1                                                                                                                                                        V                                              i                        ⁢                                                                                                  ⁢                        2                                                                                                        ]                                                          (        12        )            and
                                          {                          sI              -                                                (                                      -                                          1                      C                                                        )                                ⁡                                  [                                                                                                                                          g                            m                                                    +                                                      3                            ⁢                                                          g                              o                                                                                                                                                                            g                          m                                                                                                                                                              g                          m                                                                                                                                                  g                            m                                                    +                                                      3                            ⁢                                                          g                              o                                                                                                                                                            ]                                                      }                    ⁡                      [                                                                                V                                          o                      ⁢                                                                                          ⁢                      1                                                                                                                                        V                                          o                      ⁢                                                                                          ⁢                      2                                                                                            ]                          =                  -                                                                      g                  m                                C                            ⁡                              [                                                                            1                                                              0                                                                                                  0                                                              1                                                                      ]                                      ⁡                          [                                                                                          V                                              i                        ⁢                                                                                                  ⁢                        1                                                                                                                                                        V                                              i                        ⁢                                                                                                  ⁢                        2                                                                                                        ]                                                          (        13        )            
When Vi1=Vi2=0 (homogeneous), Equation 13 is changed into Equation 14:(sI−A)Vo=0  (14)AVo=sVo  (15)
Now we encounter an issue of an eigenvalue/eigenvector in the form of A{right arrow over (x)}=λ{right arrow over (x)}, which is common in quantum mechanics. An eigenvalue/eigenvector of the matrix A must be obtained to find a transformation matrix for diagonalizing the matrix A. The eigenvalue can be obtained by setting the determinant of the coefficient matrix at the left side of Equation 14 to 0. Since A has a form like A+=A, we see that A is a Hermitian or a self-adjoint operator. Accordingly, from the general characteristic of the Hermitian operator, the eigenvalues are expected to have real numbers and the eigenvectors are expected to be orthogonal. Actually, the eigenvalues are found to be real numbers:S=S1,S2  (16),where
      S    1    =                    -                              3            ⁢                          g              o                                C                    ⁢                          ⁢      and      ⁢                          ⁢              S        2              =          -                                    (                                          2                ⁢                                  g                  m                                            +                              3                ⁢                                  g                  o                                                      )                    C                .            
Since
            g      o        =          1              r        out              ,s1 and s2 have a dimension of a reciprocal of a time constant in an RC circuit. Corresponding eigenvectors may be obtained by applying s1 and s2 to Equation 14. The magnitudes of the eigenvectors are normalized into 1, resulting in Equations 17 and 18:
                              |                      r            1                          =                              1                          2                                ⁢                      (                                                            1                                                                                                  -                    1                                                                        )                                              (        17        )            and
                                          |                          r              2                                =                                    1                              2                                      ⁢                          (                                                                    1                                                                                        1                                                              )                                      ,                            (        18        )            where |ri> denotes a column vector and <rj| denotes a row vector.
Since the two unit vectors have a relationship of <ri|rj>=δij, they are orthogonal, as mentioned above.
Now, a transformation matrix for diagonalizing the matrix A is constructed as in Equation 19:
                    R        =                              (                                                                                                                        r                      1                                        >                                                                    ⁢                                  r                  2                                            >                        )                    =                                    1                              2                                      ⁡                          [                                                                    1                                                        1                                                                                                              -                      1                                                                            1                                                              ]                                                          (        19        )            
The matrix A is diagonalized into a diagonal matrix A by applying the transformation matrix:
                                                                                          R                  *                                ⁢                AR                            =                            ⁢                                                R                  T                                ⁢                AR                                                                                        =                            ⁢                                                R                  T                                ⁢                                  A                  ⁡                                      (                                                                                                                                                                                r                              1                                                        >                                                                                                    ⁢                                                  r                          2                                                                    >                                        )                                                                                                                          =                            ⁢                                                R                  T                                ⁡                                  (                                                                                    s                        1                                            ⁢                                                                                                                            r                            1                                                    >                                                      s                            2                                                                                                                      ⁢                                              r                        2                                                              >                                    )                                                                                                        =                            ⁢                              [                                                                                                    s                        1                                                                                    0                                                                                                  0                                                                                      s                        2                                                                                            ]                                                                                                        ≡                                ⁢                Λ                            ,                                                          (        20        )            where R+ is the transpose matrix of the complex-conjugate matrix of R. Equation 13 may be expressed as a matrix equation:
                                          (                          sI              -              A                        )                    ⁢                                    V              o                        →                          =                              (                          -                                                g                  m                                C                                      )                    ⁢                                    V              i                        →                                              (        21        )            
Performing the following linear transformation on Equation 21:{right arrow over (V)}k=R{right arrow over (V′)}k where k=i,o  (22),
We obtain Equation 23:
                                          (                          sI              -              A                        )                    ⁢          R          ⁢                                    V              o              ′                        →                          =                              (                          -                                                g                  m                                C                                      )                    ⁢                                    V              i              ′                        →                                              (        23        )            
Both sides of Equation 23 are multiplied by R−1=R+ to obtain Equations 24 to 26:
                                          R                          -              1                                ⁢                      {                          sI              -              A                        }                    ⁢          R          ⁢                                          ⁢                                    V              o              ′                        →                          =                              (                          -                                                g                  m                                C                                      )                    ⁢                                    V              i              ′                        →                                              (        24        )                                                      {                          sI              -                                                R                                      -                    1                                                  ⁢                AR                                      }                    ⁢                                    V              o              ′                        →                          =                              (                          -                                                g                  m                                C                                      )                    ⁢                                    V              i              ′                        →                                              (        25        )            and
                                          (                          sI              -              A                        )                    ⁢          R          ⁢                                    V              o              ′                        →                          =                              (                          -                                                g                  m                                C                                      )                    ⁢                                    V              i              ′                        →                                              (        26        )            
With elements of the matrices of Equation 26 shown:
                                          [                                                                                s                    -                                          s                      1                                                                                        0                                                                              0                                                                      s                    -                                          s                      2                                                                                            ]                    ⁡                      [                                                                                V                                          o                      ⁢                                                                                          ⁢                      1                                        ′                                                                                                                    V                                          o                      ⁢                                                                                          ⁢                      2                                        ′                                                                        ]                          =                  -                                                                      g                  m                                C                            ⁡                              [                                                                            1                                                              0                                                                                                  0                                                              1                                                                      ]                                      ⁡                          [                                                                                          V                                              i                        ⁢                                                                                                  ⁢                        1                                            ′                                                                                                                                  V                                              i                        ⁢                                                                                                  ⁢                        2                                            ′                                                                                  ]                                                          (        27        )            
Here, we find that the form of the transformation matrix R is exactly the same as that of the matrix describing a rotation (a rotational angle=45°) of a coordinate system on a plane. A relation of the transformation matrix R with rotational transformation will now be analyzed. For this purpose, consider the problem of a general coordinate system rotation on a plane with reference to FIG. 3. Let the matrix A represent a vector rotation which maps a vector {right arrow over (r)} into a new vector A{right arrow over (r)}:{right arrow over (r1)}=A{right arrow over (r)}  (28)
Then, a rotation of a coordinate axis B ({right arrow over (r′)}=B{right arrow over (r)}) is applied to map a given vector expression from
            [                                    x                                                y                              ]        ⁢                  ⁢          to      ⁢                          [                                                  x              ′                                                                          y              ′                                          ]        :
                                                                                             B                  ⁢                                                            r                      1                                        →                                                  =                                  BA                  ⁢                                      r                    →                                                                                                                          =                                                      BA                    ⁡                                          (                                                                        B                                                      -                            1                                                                          ⁢                        B                                            )                                                        ⁢                                      r                    →                                                                                                                          =                                                      (                                          BAB                                              -                        1                                                              )                                    ⁢                                      (                                          B                      ⁢                                              r                        →                                                              )                                                                                                            (          29          )                    
Noting that B{right arrow over (r1)} in the left side is an expression of {right arrow over (r1)} in new coordinate system and B{right arrow over (r)} in the right side is an expression of {right arrow over (r)} in new coordinate system, we know that the linear transformation denoted by matrix A in old coordinate system is now denoted by matrix A′=(BAB−1) in the new coordinate system. This means that the role played by matrix A in vector space
         [                            x                                      y                      ]  is played by matrix A′=(BAB−1) in vector space
      [                                        x            ′                                                            y            ′                                ]    .(This transformation relation is called “similarity transformation”). And when matrix B denotes a coordinate-system rotation of angle θ in a counter-clockwise direction, B is described as:
                    B        =                  [                                                                      cos                  ⁢                                                                          ⁢                  θ                                                                              sin                  ⁢                                                                          ⁢                  θ                                                                                                                          -                    sin                                    ⁢                                                                          ⁢                  θ                                                                              cos                  ⁢                                                                          ⁢                  θ                                                              ]                                    (        30        )            In this equation element bij of matrix B means a direction cosine:
                                                                                             b                  ij                                =                                ⁢                                  cos                  ⁡                                      (                                                                  x                        i                        ′                                            ·                                              x                        j                                                              )                                                                                                                          =                                ⁢                                                                            x                      ⋒                                        i                    ′                                    ·                                                            x                      ⋒                                        j                                                                                                            (          31          )                    
Now the correlation between the transformation matrix R and the matrix describing a rotation of coordinate system is described:
We can regard a vector
            V      →        k    =      [                                        V                          k              ⁢                                                          ⁢              1                                                                        V                          k              ⁢                                                          ⁢              2                                            ]  as a component representation in a reference coordinate system
      [                            x                                      y                      ]    :
                                                                                                              V                    →                                    k                                =                                ⁢                                  [                                                                                                              V                                                      k                            ⁢                                                                                                                  ⁢                            1                                                                                                                                                                                        V                                                      k                            ⁢                                                                                                                  ⁢                            2                                                                                                                                ]                                                                                                        =                                ⁢                                                                            V                                              k                        ⁢                                                                                                  ⁢                        1                                                              ⁡                                          [                                                                                                    1                                                                                                                                0                                                                                              ]                                                        +                                                            V                                              k                        ⁢                                                                                                  ⁢                        2                                                              ⁡                                          [                                                                                                    0                                                                                                                                1                                                                                              ]                                                                                                                                              =                                ⁢                                                                            V                                              k                        ⁢                                                                                                  ⁢                        1                                                              ⁢                                                                  e                        ⋒                                            1                                                        +                                                            V                                              k                        ⁢                                                                                                  ⁢                        2                                                              ⁢                                                                  e                        ⋒                                            2                                                                                                          ⁢                                  ⁢                                            where              ⁢                                                          ⁢              k                        =            i                    ,          o                                    (        32        )            
If we apply to an arbitrary fixed vector {right arrow over (V)}k a linear transformation B which represent a rotation of coordinate system from
      [                            x                                      y                      ]    ⁢          ⁢      to    ⁢                  [                                        x            ′                                                            y            ′                                ]  expression {right arrow over (Vk′)} of {right arrow over (V)}k in the new coordinate system
         [                                        x            ′                                                            y            ′                                ]  becomes:{right arrow over (V′)}k=B{right arrow over (V)}k  (33)
When we compare the form of Equation 25 with that of Equation 29, we can understand {right arrow over (V)}k′ in Equation 33 as the representation of some fixed vector in the new coordinate system after a rotation of coordinate system. If we adopt a point of view in which matrix A is regarded as an operator, it is understood that the role played by matrix A in a relation with the vector {right arrow over (V)}o in the old coordinate system is played by matrix A′=BAB−1) in a relation with the vector {right arrow over (Vo)}′ in the new coordinate system. Therefore, we find that Equation 25 is a representation of the original Equation 21 in the new coordinate system
                            [                                                            x                ′                                                                                        y                ′                                                    ]            .      Now, matrix B which describes a rotation of coordinate system needs to be found.
From Equation 22, {right arrow over (V′)}k=R−1{right arrow over (V)}k where k=i,o is obtained. When we compare this with Equation 33, we find that the matrix B which describes a rotation of coordinate system is B=R−1. Therefore, the matrix R+=RT=R−1 can be regarded as the matrix B which describes a rotation of coordinate system in a plane (Accordingly, R=B−1). In other words, if we select specially the inverse R−1 of the matrix R consisting of eigenvectors for the matrix B which describes a rotation of the coordinate system as shown in FIG. 3, the expression A′=(BAB−1) of matrix A in the new coordinate system becomes simply diagonalized as shown in Equation 27. Here,
                              R          +                =                              R            T                    =                                    1                              2                                      ⁡                          [                                                                    1                                                                              -                      1                                                                                                            1                                                        1                                                              ]                                                          (        34        )            comprises two unit row vectors <r1| and <r2|, each of which represents direction cosine associated with axes x′ and y′ respectively in the original coordinate system. Equating Equation 34 to Equation 30, we know that the angle of coordinate system rotation is given as θ=−45°. Since matrix R+=RT rotates a coordinate system to another coordinate system in which A is described as a diagonal matrix, this new coordinate system is represented as two eigenvectors |r1>, |r2>. These two eigenvectors are the unit vectors (ê1, ê2 in FIG. 4) on the new coordinate axes (called “Principal Axis”) in which matrix A is described as a diagonal matrix.
Now apply the following linear transformation to input signal pair and output signal pair using the transformation matrix R derived above (For convenience, we use DMk instead of V′k1 and CMk instead of V′k2:
                                                                                                                                    [                                                                                                                                  V                                                              k                                ⁢                                                                                                                                  ⁢                                1                                                                                                                                                                                                                        V                                                              k                                ⁢                                                                                                                                  ⁢                                2                                                                                                                                                        ]                                        =                                                                                            1                                                      2                                                                          ⁡                                                  [                                                                                                                    1                                                                                            1                                                                                                                                                                                      -                                  1                                                                                                                            1                                                                                                              ]                                                                    ⁡                                              [                                                                                                                                            D                                ⁢                                                                                                                                  ⁢                                                                  M                                  k                                                                                                                                                                                                                                        C                                ⁢                                                                                                                                  ⁢                                                                  M                                  k                                                                                                                                                                    ]                                                                              ︸                                ⁢                                                                              R                        ⁢            where            ⁢                                                  ⁢            k                    =          i                ,        o                            (        35        )            
From this, we obtain:
                                          [                                                                                D                    ⁢                                                                                  ⁢                                          M                      k                                                                                                                                        C                    ⁢                                                                                  ⁢                                          M                      k                                                                                            ]                    =                                                                                          1                                          2                                                        ⁡                                      [                                                                                            1                                                                                                      -                            1                                                                                                                                                1                                                                          1                                                                                      ]                                                  ⁡                                  [                                                                                                              V                                                      k                            ⁢                                                                                                                  ⁢                            1                                                                                                                                                                                        V                                                      k                            ⁢                                                                                                                  ⁢                            2                                                                                                                                ]                                            ⁢                                                          ⁢              where              ⁢                                                          ⁢              k                        =            i                          ,        o                            (        36        )            
From Equation 36, the following equations are obtained:
                              [                                                                      V                                      o                    ⁢                                                                                  ⁢                    1                                                                                                                        V                                      o                    ⁢                                                                                  ⁢                    2                                                                                ]                =                                                            1                                  2                                            ⁡                              [                                                                            1                                                              1                                                                                                                          -                        1                                                                                    1                                                                      ]                                      ⁡                          [                                                                                          D                      ⁢                                                                                          ⁢                                              M                        o                                                                                                                                                        C                      ⁢                                                                                          ⁢                                              M                        o                                                                                                        ]                                ⁢                                          ⁢                      (                          k              =              o                        )                                              (        37        )            and
                              [                                                                      V                                      i                    ⁢                                                                                  ⁢                    1                                                                                                                        V                                      i                    ⁢                                                                                  ⁢                    2                                                                                ]                =                                                            1                                  2                                            ⁡                              [                                                                            1                                                              1                                                                                                                          -                        1                                                                                    1                                                                      ]                                      ⁡                          [                                                                                          D                      ⁢                                                                                          ⁢                                              M                        i                                                                                                                                                        C                      ⁢                                                                                          ⁢                                              M                        i                                                                                                        ]                                ⁢                                          ⁢                      (                          k              =              i                        )                                              (        38        )            If we insert Equations 37 and 38 to Equation 13 and arrange the resultant equation, we obtain:
                              [                                                                      D                  ⁢                                                                          ⁢                                      M                    o                                                                                                                        C                  ⁢                                                                          ⁢                                      M                    o                                                                                ]                =                                            g              m                        ⁡                          [                                                                                                                  -                        1                                                                    sC                        +                                                  3                          ⁢                                                      g                            o                                                                                                                                                    0                                                                                        0                                                                                                      -                        1                                                                    sC                        +                                                  2                          ⁢                                                      g                            m                                                                          +                                                  3                          ⁢                                                      g                            o                                                                                                                                                          ]                                ⁡                      [                                                                                D                    ⁢                                                                                  ⁢                                          M                      i                                                                                                                                        C                    ⁢                                                                                  ⁢                                          M                      i                                                                                            ]                                              (        39        )            
Physical amplifications which Equation 39 has are as follows: Equation 39 represent forward-direction transfer characteristic from input DMi, CMi at input port to output DMo, CMo at output port. First of all, we find in this transfer characteristic that common-mode of input signal CMi has influence only on common-mode of output signal CMo and that differential-mode of input signal DMi has influence only on differential-mode of output signal DMo. This means that decoupling of differential-mode from common-mode is perfectly achieved.
This characteristic is very desirable, considering that the signal we want to process is not contained in the common-mode. Furthermore, the transconductor will approach the more ideal characteristic if common-mode signal gain can be completely nullified. However, we can point out that it is not easy to increase only the quality factor of the conventional Nauta-OTA, as is seen in equation 39.
In fact, in case of UWB transceiver where very high frequency operation in the range of several hundreds of MHz is required, gm value needed is very large so that the quality factor becomes substantially reduced. Nevertheless, any practical means independently to control the quality factor is not provided, raising an important issue. The following OTA structure is proposed to resolve these problems of the conventional Nauta-OTA.